Tic Tac Toe Carnival Game

A recent carnival was offering a tic tac toe style game. For £1 a go you throw three incredibly bouncy balls towards a large wooden box with 9 pockets in the bottom. The front face had clear perspex on the front so you can see where the balls landed.

I had a go and to be able to aim is nigh on impossible, so I figured the chances are purely random. The prize was either a massive cuddly toy, or £20. I was obviously going for the £20 ;-)

I spent £5 before giving up.

I have now attempted to figure the odds, assuming random placement.

I believe there are three distinct patterns.

1) The first ball lands in the middle, the second ball can land anywhere, the third has one slot to hit. So i figure 1/9 * 8/8 * 1/7 = 1/63
2) The first ball lands on a corner, the second ball has 4 possible pockets, the third again has one. So 1/9 * 4/8 * 1/7 = 1/126
3) The ball lands on the outer middle, but has exactly the same odds as above.

I think I now combine these odds, and get a probability of 1 in 31.5

Is this right?

How many different ways can 3 the balls land? There are 9 slots, so we get 9*8*7 = 504
How many different ways cause a win? You have to count them by hand:

3 of this style
x.x.x
o.o.o
o.o.o

3 of this style

x.o.o
x.o.o
x.o.o

and 2 of this style

x.o.o
o.x.o
o.o.x

So 8 ways, and the ball can form one of these 8 patterns in 6 ways (123,132,213,231,312,321) 8*6 = 48

Answer 48/504 = 1 in 10.5

Of course, this assumes there isn't some sticky tape in a hole, or some other dishonest trick!

Pretty sure I was able to get a ball in every hole, but I am sure it was £20, so it that respect it was a good bet, but I watched for ages before having a go and no-one won. Hmmm

Quote: WizardofEngland

2) The first ball lands on a corner, the second ball has 4 possible pockets, the third again has one. So 1/9 * 4/8 * 1/7 = 1/126

If the first ball lands in a corner, there are SIX good pockets for the second ball.


But I agree with CardShark: The overall odds really depend on there being equal odds for any available pocket on any throw. Carnival games, almost by definition, can be counted on to NOT be played fairly.

Quote: DJTeddyBear

If the first ball lands in a corner, there are SIX good pockets for the second ball.


But I agree with CardShark: The overall odds really depend on there being equal odds for any available pocket on any throw. Carnival games, almost by definition, can be counted on to NOT be played fairly.

Your very right, don't know how I missed that

Usually with tossing balls into a container its rigged to bounce out. Often if the game uses smaller balls the Carney will place one of them inside the container and then toss another ball inside to prove that it will stay there, if tossed. But its that first ball that is the one likely to bounce out.

If they use larger balls then its often that from the angle you can see all containers are the same but in actuality the valuable container will have a bottom that encourages a bounce.

Here is how I would solve it. There are 8 ways to win (three rows, three columns, and two diagonals). There are combin(9,3)=84 possible combinations of 3 squares out of 9. 8/84 = 1 in 10.5.

So since wiz and card shark figured out if it is fair then the carnie will be losing money hand over fist. Most likely is the center hole is then least likely to accept a ball and the 4 non corners the most likely. This will skew the odds in the carnies favor significantly. Good for you that you only spent 5 pounds for the lesson.

Quote: SOOPOO

So since wiz and card shark figured out if it is fair then the carnie will be losing money hand over fist. Most likely is the center hole is then least likely to accept a ball and the 4 non corners the most likely. This will skew the odds in the carnies favor significantly. Good for you that you only spent 5 pounds for the lesson.

I wonder how skewed they differant holes have to be? Excel here I come.

Quote: SOOPOO

So since wiz and card shark figured out if it is fair then the carnie will be losing money hand over fist. Most likely is the center hole is then least likely to accept a ball and the 4 non corners the most likely. This will skew the odds in the carnies favor significantly. Good for you that you only spent 5 pounds for the lesson.

You have to be in the spirit for a carnival though, and then there is the macho factor, trying to look good by winning, but ending up a loser.